I have a prediction f ( x ) of some continuous process variable, based on an input va

Llubanipo

Llubanipo

Answered question

2022-06-15

I have a prediction f ( x ) of some continuous process variable, based on an input variable x (think: location). The prediction is incorrect, with the error being normal distributed with expected value μ and standard deviation σ.
Hence, the probability density function of f ( x ) should be
1 σ 2 π e ( ( f ( x ) μ ) / σ ) 2 2
Is this correct? (No it is not, see answer below)
Now, I have a measurement m of the process variable, based on an unknown input variable x m m is assumed to be correct, but quantized to integral numbers.
Given a set of x i together with their predictions f ( x i ), how can I compute a probability that x m was in the vicinity of x i ?
I apologize if the question makes no sense.

Answer & Explanation

laure6237ma

laure6237ma

Beginner2022-06-16Added 27 answers

Your proposed density function is not correct. Rather, the probability density function of the error ε is
w 1 σ 2 π e ( ( w μ ) / σ ) 2 2 .
If the prediction is f ( x ), and the true value of the thing to be predicted is g ( x ), then
f ( x ) = g ( x ) + ε .
You will never have an exact match since the probability distribution of the error has no discrete component---it's a continuous distribution. So the probability of an exact match is 0. But you can speak of the probability that the prediction differs by no more than a specified amount from the true value. That probability would depend on μ and σ.

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